Simplify the following expression: $x = \dfrac{8q^3 + 24q^2}{-32q^3 - 4q^2}$ You can assume $q \neq 0$.
Find the greatest common factor of the numerator and denominator. The numerator can be factored: $8q^3 + 24q^2 = (2\cdot2\cdot2 \cdot q \cdot q \cdot q) + (2\cdot2\cdot2\cdot3 \cdot q \cdot q)$ The denominator can be factored: $-32q^3 - 4q^2 = - (2\cdot2\cdot2\cdot2\cdot2 \cdot q \cdot q \cdot q) - (2\cdot2 \cdot q \cdot q)$ The greatest common factor of all the terms is $4q^2$ Factoring out $4q^2$ gives us: $x = \dfrac{(4q^2)(2q + 6)}{(4q^2)(-8q - 1)}$ Dividing both the numerator and denominator by $4q^2$ gives: $x = \dfrac{2q + 6}{-8q - 1}$